16 ON THE SECTIONS OF A BLOCK OF EIGHTCELLS 



the middle layer on the middle strip of fig. 8 we find that the 

 projection of any of these eightcells covers a third pari; of the 

 strip, consisting of three of the nine parts into which this strip 

 can be divided by lines parallel to P Q - Now, if we arrange 

 (fig. 0) the 27 cubes in groups, and reckon as belonging to the 

 same group these cubes that cover in projection on A A' the same 

 third part, we alight evidently on the seven groups found above. 

 So in order to apply this result to the projection of the block of 

 eightcells, we have repeated under the projection (fig. 8) the line 

 A A' in seven parallel positions A l A l ', A 2 A 2 ',. . ., A^A^ ', indicated 

 on each of them the position of the third part that forms the pro- 

 jection and inscribed at each of these third parts the numbers 



19; 10, 20, 22; 1, 11, 13, 21, 23, 25; etc. 

 of the eightcells projecting themselves on the corresponding third 

 part of the strip. For convenience we will designate these groups 

 of eightcells by G x , G s , G 6 , G^, G 6 ', G 3 ', G x ' indicating by the 

 subscript the number of the eightcells, and we will add the 

 signs -J- and — to the corresponding groups of the plus layer and 

 the minus layer 1 ). 



13. Now we are cpiite provided with the means of determining, 

 for any arbitrary position of the intersecting space S^jr), the 

 number, form and size of the different solids, sections of separate 

 eightcells which fill up the section of the large block of eightcells, 

 which is itself an eightcell of three times the size. We may even 

 assert that this process, complicated as it seems, in reality is 

 easier than another to which we are accustomed and which we 

 perforin daily: "to see what o'clock it is." For our dial — the 

 plane t' on to which the block of eightcells has been projected — 

 has one hand only — the projection of the intersecting space 

 turning round 0. But in order to facilitate the enumeration of 

 the results it will be well to introduce beforehand a simple nota- 

 tion for the different kinds of solids we obtain. 



As we have seen the section of any of the 81 eightcells is a 



') If we replace the 27 cubes by their centres, the projection of these centres on a 

 diagonal of the block of cubes gives us a range of seven points coinciding with the 

 centres of the rectangles P U PQQ , P'P 'Q 'Q' and the five .points dividing the segment 

 of A A' limited at these two points into six equal parts. So the numbers 1, 3, 6, 7, 

 6, 3, 1 are found — compare the preceding note — as the coefficients of the powers 

 of x in (1 -f- x -f~ x 1 ) 3 . 



In the extension of the problem to the measure-polytope ƒ>., /( U') with edges equal to 

 p units of space S n divided into p n measure-poly topes P-^n^ with edges unity, the 

 numbers of polytopes of the groups G coinciding in projection on w' (compare the pre- 

 ceding note) are the coefficients of the powers of x in (1 -\- x -f-£C* + . . . . -\-xi'-' 1 )". 



